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Freely 2-periodic knots have two canonical components

Keegan Boyle, Nicholas Rouse

Abstract

We prove that the SL(2, C) character variety of a hyperbolic, freely 2-periodic knot has two canonical components. We also prove that the hyperbolic torsion polynomial of such a knot satisfies a factorization condition which seems to be particularly effective at identifying freely 2-periodic knots.

Freely 2-periodic knots have two canonical components

Abstract

We prove that the SL(2, C) character variety of a hyperbolic, freely 2-periodic knot has two canonical components. We also prove that the hyperbolic torsion polynomial of such a knot satisfies a factorization condition which seems to be particularly effective at identifying freely 2-periodic knots.
Paper Structure (5 sections, 7 theorems, 7 equations, 1 figure)

This paper contains 5 sections, 7 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Questions
  3. Acknowledgments
  4. Proof of Theorem \ref{['thm:main']}
  5. Hyperbolic torsion for freely 2-periodic knots

Key Result

Theorem 1.1

If $K \subseteq S^3$ is a hyperbolic, freely $2$-periodic knot, then the $\mathrm{SL}({2},{\mathbf{C}})$ character variety of $\pi_1(S^3 \setminus K)$ has two canonical components.

Figures (1)

  • Figure 1: The freely 2-periodic knot $10_{157}$. The symmetry is the composition of $\pi$ rotations around two disjoint circles, indicated by the two arrows. By Theorem \ref{['thm:main']}$X(10_{157})$ has two canonical components.

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 11 more